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Theorem maduval 18900
Description: Second substitution for the adjunct (cofactor) matrix. (Contributed by SO, 11-Jul-2018.)
Hypotheses
Ref Expression
madufval.a  |-  A  =  ( N Mat  R )
madufval.d  |-  D  =  ( N maDet  R )
madufval.j  |-  J  =  ( N maAdju  R )
madufval.b  |-  B  =  ( Base `  A
)
madufval.o  |-  .1.  =  ( 1r `  R )
madufval.z  |-  .0.  =  ( 0g `  R )
Assertion
Ref Expression
maduval  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
Distinct variable groups:    i, N, j, k, l    R, i, j, k, l    i, M, j, k, l
Allowed substitution hints:    A( i, j, k, l)    B( i, j, k, l)    D( i, j, k, l)    .1. ( i, j, k, l)    J( i, j, k, l)    .0. ( i, j, k, l)

Proof of Theorem maduval
Dummy variable  m is distinct from all other variables.
StepHypRef Expression
1 madufval.a . . . . 5  |-  A  =  ( N Mat  R )
2 madufval.b . . . . 5  |-  B  =  ( Base `  A
)
31, 2matrcl 18674 . . . 4  |-  ( M  e.  B  ->  ( N  e.  Fin  /\  R  e.  _V ) )
43simpld 459 . . 3  |-  ( M  e.  B  ->  N  e.  Fin )
5 eqid 2460 . . . 4  |-  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
65mpt2exg 6848 . . 3  |-  ( ( N  e.  Fin  /\  N  e.  Fin )  ->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e.  _V )
74, 4, 6syl2anc 661 . 2  |-  ( M  e.  B  ->  (
i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e. 
_V )
8 oveq 6281 . . . . . . . . 9  |-  ( m  =  M  ->  (
k m l )  =  ( k M l ) )
98ifeq2d 3951 . . . . . . . 8  |-  ( m  =  M  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
1093ad2ant1 1012 . . . . . . 7  |-  ( ( m  =  M  /\  k  e.  N  /\  l  e.  N )  ->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) )  =  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) )
1110mpt2eq3dva 6336 . . . . . 6  |-  ( m  =  M  ->  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) )  =  ( k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
12113ad2ant1 1012 . . . . 5  |-  ( ( m  =  M  /\  i  e.  N  /\  j  e.  N )  ->  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) )  =  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) )
1312fveq2d 5861 . . . 4  |-  ( ( m  =  M  /\  i  e.  N  /\  j  e.  N )  ->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) )  =  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )
1413mpt2eq3dva 6336 . . 3  |-  ( m  =  M  ->  (
i  e.  N , 
j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
15 madufval.d . . . 4  |-  D  =  ( N maDet  R )
16 madufval.j . . . 4  |-  J  =  ( N maAdju  R )
17 madufval.o . . . 4  |-  .1.  =  ( 1r `  R )
18 madufval.z . . . 4  |-  .0.  =  ( 0g `  R )
191, 15, 16, 2, 17, 18madufval 18899 . . 3  |-  J  =  ( m  e.  B  |->  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k m l ) ) ) ) ) )
2014, 19fvmptg 5939 . 2  |-  ( ( M  e.  B  /\  ( i  e.  N ,  j  e.  N  |->  ( D `  (
k  e.  N , 
l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) )  e.  _V )  -> 
( J `  M
)  =  ( i  e.  N ,  j  e.  N  |->  ( D `
 ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
217, 20mpdan 668 1  |-  ( M  e.  B  ->  ( J `  M )  =  ( i  e.  N ,  j  e.  N  |->  ( D `  ( k  e.  N ,  l  e.  N  |->  if ( k  =  j ,  if ( l  =  i ,  .1.  ,  .0.  ) ,  ( k M l ) ) ) ) ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ w3a 968    = wceq 1374    e. wcel 1762   _Vcvv 3106   ifcif 3932   ` cfv 5579  (class class class)co 6275    |-> cmpt2 6277   Fincfn 7506   Basecbs 14479   0gc0g 14684   1rcur 16936   Mat cmat 18669   maDet cmdat 18846   maAdju cmadu 18894
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-rep 4551  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-csb 3429  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-op 4027  df-uni 4239  df-iun 4320  df-br 4441  df-opab 4499  df-mpt 4500  df-id 4788  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-1st 6774  df-2nd 6775  df-slot 14483  df-base 14484  df-mat 18670  df-madu 18896
This theorem is referenced by:  maducoeval  18901
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